What you are asking for is called a proxy http server.
The software receives http requests and passes them on or rejects them (logging too) It needn't sit on your embeded device, however your device would need to be configured to access the proxy instead of going directly to the internet.
nginx are two examples of web servers that can act as proxy servers.
squid is a sophisticated proxy server which can handle many protocols.
charles is another, but paid for program (although there is a free trial period).
If you need to ensure that all requests are via the proxy then you would also want to set up your network to prevent direct requests from the device going directly to the internet. (you can't do that if your device hosts the proxy).